(Existence of adapted wavelets) For
each
we define the functions
:
Then

1.
are linearly independent,

2.
,

3.
are
-orthogonal
to all
.

Proof

The claim 1 follows by contradiction. The collection
is a part of a finite basis for
.
The dimension of
increases by factor of
when increasing
by
.
Linear dependence of
would mean that
(and thus
)
is missing a dimension. This contradicts the proposition
(
Maximal dimension 1
).

According to the formula
,
According to the construction of
,
the functions
are orthogonal to
for every
.
Hence,
This is the claim 2.